Constructing class fields over local fields
Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 627-652.

Soit K un corps 𝔭-adique. Nous donnons une caractérisation explicite des extensions abéliennes de K de degré p en reliant les coefficients des polynômes engendrant les extensions L/K de degré p aux exposants des générateurs du groupe des normes N L/K (L * ). Ceci est appliqué à un algorithme de construction des corps de classes de degré p m , ce qui conduit à un algorithme de calcul des corps de classes en général.

Let K be a 𝔭-adic field. We give an explicit characterization of the abelian extensions of K of degree p by relating the coefficients of the generating polynomials of extensions L/K of degree p to the exponents of generators of the norm group N L/K (L * ). This is applied in an algorithm for the construction of class fields of degree p m , which yields an algorithm for the computation of class fields in general.

@article{JTNB_2006__18_3_627_0,
     author = {Pauli, Sebastian},
     title = {Constructing class fields over local fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {627--652},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {3},
     year = {2006},
     doi = {10.5802/jtnb.563},
     mrnumber = {2330432},
     zbl = {1136.11072},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.563/}
}
Pauli, Sebastian. Constructing class fields over local fields. Journal de Théorie des Nombres de Bordeaux, Tome 18 (2006) no. 3, pp. 627-652. doi : 10.5802/jtnb.563. http://www.numdam.org/articles/10.5802/jtnb.563/

[Ama71] S. Amano, Eisenstein equations of degree p in a 𝔭-adic field. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 1–21. | MR 308086 | Zbl 0231.12019

[BC95] W. Bosma, J.J. Cannon, Handbook of Magma functions. School of Mathematics, University of Sydney, Sydney, 1995.

[Coh99] H. Cohen, Advanced topics in computational number theory. Springer Verlag, New York, 1999. | MR 1728313 | Zbl 0977.11056

[Fie99] C. Fieker, Computing class fields via the Artin map. Math. Comp. 70 (2001), 1293–1303. | MR 1826583 | Zbl 0982.11074

[FV93] I. B. Fesenko, S. V. Vostokov, Local fields and their extensions. Translations of Mathematical Monographs, vol. 121, American Mathematical Society, 1993. | MR 1218392 | Zbl 0781.11042

[Has80] H. Hasse, Number Theory. Springer Verlag, Berlin, 1980. | MR 562104 | Zbl 0423.12002

[HPP03] F. Hess, S. Pauli, M. E. Pohst, Computing the multiplicative group of residue class rings. Math. Comp. 72 (2003), no. 243, 1531–1548. | MR 1972751 | Zbl 1013.11073

[Iwa86] K. Iwasawa, Local class field theory. Oxford University Press, New York, 1986. | MR 863740 | Zbl 0604.12014

[Kra66] M. Krasner, Nombre des extensions d’un degré donné d’un corps 𝔭-adique Les Tendances Géométriques en Algèbre et Théorie des Nombres, Paris, 1966, 143–169. | Zbl 0143.06403

[MW56] R. E. MacKenzie, G. Whaples, Artin-Schreier equations in characteristic zero. Amer. J. Math. 78 (1956), 473–485. MR 19,834c | MR 90584 | Zbl 0073.26402

[Pan95] P. Panayi, Computation of Leopoldt’s p-adic regulator. Dissertation, University of East Anglia, 1995.

[PR01] S. Pauli, X.-F. Roblot, On the computation of all extensions of a p-adic field of a given degree. Math. Comp. 70 (2001), 1641–1659. | MR 1836924 | Zbl 0981.11038

[Ser63] J.-P. Serre, Corps locaux. Hermann, Paris, 1963. | MR 354618 | Zbl 0137.02601

[Yam58] K. Yamamoto, Isomorphism theorem in the local class field theory. Mem. Fac. Sci. Kyushu Ser. A 12 (1958), 67–103. | MR 150136 | Zbl 0083.25901